### Fractions of a Cycle

Engineers break the ac cycle down into small parts for analysis and reference. One complete cycle can be compared to a single revolution around a circle. Degrees One method of specifying the phase of an ac cycle is to divide it into 360 equal parts, called degrees or degrees of phase, symbolized by a superscript, lowercase letter o (°). The value 0° is assigned to the point in the cycle where the magnitude is zero and positive-going. The same point on the next cycle is given the value 360°. The point one-fourth of the way through the cycle is 90°; the point halfway through the cycle is 180°; the point three-fourths of the way through the cycle is 270°. This is illustrated in Above Figure. Degrees of phase are used mainly by engineers and technicians.

#### Radians

The other method of specifying phase is to divide the cycle into 2π equal parts, where π (pi) is a geometric constant equal to the number of diameters of any circle that can be laid end to end around the circumference of that circle. This constant is approximately equal to 3.14159. A radian (rad) of phase is thus equal to about 57.3°. Sometimes, the frequency of an ac wave is measured in radians per second (rad/s) rather than in hertz. Because there are about 6.28 radians in a complete cycle of 360°, the angular frequency of a wave, in radians per second, is equal to about 6.28 times the frequency in hertz. Radians of phase are used mainly by physicists.

#### Phase Difference

Even if two ac waves have exactly the same frequency, they can have different effects because they are out of sync with each other. This is especially true when ac waves are added together to produce a third, or composite, wave.

If two pure ac sine waves have identical frequencies and identical amplitudes but differ in phase by 180° (a half cycle), they cancel each other out, and the composite wave is zero; it ceases to exist! If the two waves are exactly in phase, the composite wave has the same frequency, but twice the amplitude, of either signal alone.

If two pure ac sine waves have the same frequency but different amplitudes, and if they differ in phase by 180°, the composite signal has the same frequency as the originals, and an amplitude equal to the difference between the two. If two such waves are exactly in phase, the composite has the same frequency as the originals, and an amplitude equal to the sum of the two.

If two pure ac sine waves have the same frequency but differ in phase by some odd amount such as 75° or 110°, the resulting signal has the same frequency, but does not have the same waveshape as either of the original signals. The variety of such cases is infinite.

Household electricity from 117-V wall outlets consists of a 60-Hz sine wave with only one phase component. But the energy is transmitted over long distances in three phases, each differing by 120° or one-third of a cycle. This is what is meant by three-phase ac. Each of the three ac waves carries one-third of the total power in a utility transmission line.